Carsten Timm: Theory of superconductivity

with
η = − 1 k B T β
> 0. (7.11)
4π γα
Thus the correlation function of the order parameters decays like a power law of distance in two dimensions. We do
not find long-range order, which would imply lim r→∞ ⟨ψ(r) ∗ ψ(0)⟩ = const. This agrees with the Mermin-Wagner
theorem, which forbids long-range order for the two-dimensional superfluid. The power-law decay characterizes
so-called quasi-long-range order (short range order would have an even faster, e.g., exponential, decay).
Isolated vortices
We have argued that fluctuations in the amplitude are less important because they have an energy gap proportional
to −2α > 0. This is indeed true for small amplitude fluctuations. However, there exist variations of the amplitude
that are, while energetically costly, very stable once they have been created. These are vortices. In two dimensions,
a vortex is a zero-dimensional object; the order parameter goes to zero at a single point at its center. The simplest
form of a vortex at the origin can be represented by
ψ(r) = |ψ(r)| e iϕ(r) = |ψ(r)| e i(φ−φ 0) , (7.12)
where r and φ are (planar) polar coordinates of r. An antivortex would be described by
ψ(r) = |ψ(r)| e −i(φ−φ 0) . (7.13)
In both cases, lim r→0 |ψ(r)| = 0. Note that we have changed the convention for the sign in the exponent compared
to superconducting vortices.
In the presence of vortices, the phase ϕ(r) of the order parameter is multivalued and, of course, undefined at
the vortex centers. On the other hand, the phase gradient v = ∇ϕ is single-valued (but still undefined at the
vortex centers). For any closed loop C not touching any vortex cores, we have
∮
ds · v = total change in phase along C = 2π N C , (7.14)
C
where N C ∈ Z is the enclosed winding number or vorticity. The vorticity can be written as the sum of the
vorticities N i = ±1 of all vortices and antivortices inside the loop,
N C = ∑ i
N i . (7.15)
N = +1
1
N = +1
2
N = −1
3
Defining the vortex concentration by
n v (r) := ∑ i
N i δ(r − R i ), (7.16)
where R i is the location of the center of vortex i, we obtain
∇ × v ≡ ∂
∂x v y − ∂ ∂y v x = 2π n v (r). (7.17)
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